(a) Prove that if the energy eigenvalues of a system can be expressed as a sum of independent contributions E = EA + EB +EC (e.g. electronic energy, vibrational energy, rota-tional energy) that the...

(a) Prove that if the energy eigenvalues of a system can be expressed as a sum of independent contributions E = E_A

+ E_B
+E_C

(e.g. electronic energy, vibrational energy, rota-tional energy) that the heat capaicity can be written C_V
= C_V
^(A)
+C_V
(B) +C_v
^{(C) .}
In addition, show that the heat capacity is independent of zero point energy.

(b) Derive an expression for the electronic heat capacity assuming that there are only three significant electronic states and that they have energies and degeneracies given by ε₀, g₀, ε₀, g₁, ε₂, g₂
.

(c) Given that the energies required for electronic transitions correspond roughly to u.v. light (-50,000^oK), show how the molecule will change if the electronic degrees of freedom are totally neglected. What if the ground electro-nic state degeneracy is included but all the excited electro-nic states are neglected?

(d) Show how the room temperature entropy of the same molecule will change in these two cases.